DOT project 2.ipynb

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   "metadata": {},
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   "source": [
    "%matplotlib widget\n",
    "#%matplotlib inline\n",
    "from IPython.display import display, Markdown\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.colors as mcolors\n",
    "from matplotlib import ticker\n",
    "from matplotlib import cm\n",
    "import numpy as np\n",
    "import time\n",
    "import plotly.graph_objects as go"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "## Problem data\n",
    "\n",
    "n=1 #refraction index\n",
    "c= 30/n # cm/ns\n",
    "mus= 10 #cm^-1\n",
    "mua=0.1 # cm^-1\n",
    "Dmua=0.1 #cm^-1\n",
    "D=1/(3*mus) #cm\n",
    "mm= 1e-1 # cm\n",
    "\n",
    "dt=1\n",
    "t=np.arange(0,8,dt) # ns\n",
    "\n",
    "V=1 #cm^3 perturbation dimension\n",
    "\n",
    "RP= (V*3/(4*np.pi))**(1/3) #cm perturbation radius\n",
    "\n",
    "step=5\n",
    "\n",
    "x=y=z= np.arange(0,64.1,step) * mm\n",
    "\n",
    "xs=ys=np.arange(4,60.1,step) * mm\n",
    "\n",
    "Xs, Ys,Xd,Yd = np.meshgrid(xs,ys,xs,ys) #source-detector grid\n",
    "\n",
    "Xsf=Xs.flatten()\n",
    "Ysf=Ys.flatten()\n",
    "Xdf=Xd.flatten()\n",
    "Ydf=Yd.flatten()\n",
    "\n",
    "X,Y,Z= np.meshgrid(x,y,z)  #space voxels\n",
    "\n",
    "Xf,Yf,Zf=X.flatten(), Y.flatten(), Z.flatten()\n",
    "\n",
    "xp,yp,zp = [1.5, 2.5 ,1]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Absorption perturbation vector $\\vec A$\n",
    "\n",
    "Calculate A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "mask =(X-xp)**2+(Y-yp)**2+(Z-zp)**2 <RP**2\n",
    "\n",
    "maskf =mask.flatten()\n",
    "\n",
    "A= maskf*Dmua"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n° elements of A: 2197\n",
      "dimension of A reshaped: (13, 13, 13)\n",
      "Unit of measurment of A is cm^-1\n"
     ]
    }
   ],
   "source": [
    "print(\"n° elements of A:\",A.size)\n",
    "A3D=A.reshape((x.size,y.size,z.size))\n",
    "print(\"dimension of A reshaped:\",A3D.shape)\n",
    "print(\"Unit of measurment of A is cm^-1\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "1430ed6479ee4d16b5475f5f1feccbb5",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.rcParams['figure.dpi'] = 200\n",
    "plt.rcParams[\"figure.figsize\"] = [5,5]\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = fig.gca(projection='3d')\n",
    "\n",
    "idx= A!=0\n",
    "\n",
    "p=ax.scatter(Xf[idx], Yf[idx], Zf[idx], c=A[idx],cmap=cm.viridis,s=60, marker='o')\n",
    "plt.xlim(0, 6.4)\n",
    "plt.ylim(0, 6.4)\n",
    "ax.set_zlim(0,6.4)\n",
    "plt.colorbar(p)  \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
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\"ticks\": \"\"}}, \"title\": {\"x\": 0.05}, \"xaxis\": {\"automargin\": true, \"gridcolor\": \"white\", \"linecolor\": \"white\", \"ticks\": \"\", \"title\": {\"standoff\": 15}, \"zerolinecolor\": \"white\", \"zerolinewidth\": 2}, \"yaxis\": {\"automargin\": true, \"gridcolor\": \"white\", \"linecolor\": \"white\", \"ticks\": \"\", \"title\": {\"standoff\": 15}, \"zerolinecolor\": \"white\", \"zerolinewidth\": 2}}}},\n",
       "                        {\"responsive\": true}\n",
       "                    ).then(function(){\n",
       "                            \n",
       "var gd = document.getElementById('6be13b0d-2a0c-4865-8790-3048536535a4');\n",
       "var x = new MutationObserver(function (mutations, observer) {{\n",
       "        var display = window.getComputedStyle(gd).display;\n",
       "        if (!display || display === 'none') {{\n",
       "            console.log([gd, 'removed!']);\n",
       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
       "}}\n",
       "\n",
       "                        })\n",
       "                };\n",
       "                });\n",
       "            </script>\n",
       "        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "graph= 1 # 1 o 2\n",
    "if graph == 1:\n",
    "    fffig = go.Figure(data=go.Volume(x=Xf, y=Yf, z=Zf,value=A,\n",
    "        opacity=0.1,# needs to be small to see through all surfaces\n",
    "        surface_count=50 #needs to be a large number for good volume rendering\n",
    "        ))\n",
    "    fffig.show()\n",
    "\n",
    "if graph ==2:\n",
    "    \n",
    "    fig = plt.figure()\n",
    "    xi,yi,zi = np.indices(np.array(A3D.shape)+1)/A3D.shape[1] *x[-1]\n",
    "    ax = fig.gca(projection='3d')\n",
    "    ax.voxels(xi,yi,zi,A3D,edgecolor='k')\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sensitivity Matrix $W$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9.740844488143921\n"
     ]
    }
   ],
   "source": [
    "start = time.time()\n",
    "i,m= np.ogrid[:Xsf.size,:Xf.size]\n",
    "start = time.time()\n",
    "\n",
    "rho12= np.sqrt((Xsf[i]-Xf[m])**2 + (Ysf[i]-Yf[m])**2 +(Zf[m])**2) # distance source-pert\n",
    "rho23= np.sqrt((Xdf[i]-Xf[m])**2 + (Ydf[i]-Yf[m])**2 +(Zf[m])**2) # distance detector-pert\n",
    "r=np.sqrt((Xsf[i]-Xdf[i])**2 + (Ysf[i]-Ydf[i])**2)\n",
    "\n",
    "\n",
    "W=0\n",
    "tt=(t[1:] + t[:-1]) / 2\n",
    "np.seterr(divide='ignore')\n",
    "Sp=-1/(4*np.pi*D)*dt*((step*0.1)**3)*(1/rho12 + 1/rho23)\n",
    "\n",
    "W= list(map(lambda t: Sp*np.exp((-(rho12+rho23)**2 +r**2) /(4*c*D*t)),tt))\n",
    "W=np.sum(W,axis=0)/t[-1]\n",
    "\n",
    "W[np.isinf(W)]=0\n",
    "#W=np.nan_to_num(W)\n",
    "end = time.time()\n",
    "print(end - start)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n° elements of W: 45556992\n",
      "dimension of W: (20736, 2197)\n",
      "Unit of measurment of A is cm^-1\n"
     ]
    }
   ],
   "source": [
    "print(\"n° elements of W:\",W.size)\n",
    "print(\"dimension of W:\",W.shape)\n",
    "print(\"Unit of measurment of A is cm^-1\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "6b53266a21864d359cfb405522eb03a4",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.imshow(W,cmap=cm.jet,aspect='auto')\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.875\n"
     ]
    }
   ],
   "source": [
    "V_vox=np.sum(maskf)*(step*0.1)**3\n",
    "print(V_vox)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Measurments vector M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "Mf=np.inner(W,A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "M=Mf.reshape((xs.size,ys.size,xs.size,ys.size))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [],
   "source": [
    "Msum=np.sum(M,(2,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "11b6746f77c44e5b86df81fa798a7504",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.close()\n",
    "plt.figure()\n",
    "plt.imshow(Msum,cmap='jet',aspect='equal',extent=[4,60,60,4])\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.28288976])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M[np.where(M==M.min())]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Singular values & vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "[U,s,Vh]=np.linalg.svd(W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "a15124c1b3534e81921219aff5d7c1eb",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.imshow(np.diag(s),cmap='viridis',aspect='equal')\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.imshow(U,cmap='viridis',aspect='equal')\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.close()\n",
    "plt.figure()\n",
    "Vmin=Vh.transpose()[:(min(W.shape)),:(min(W.shape))]\n",
    "plt.imshow(Vmin,cmap='viridis',aspect='equal')\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Inverse problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "S=np.diag(s)\n",
    "S.shape\n",
    "Sinv= np.zeros(W.shape).transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "i=range(s.size)\n",
    "Sinv[i,i]=1/s[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "Winv= np.linalg.multi_dot([Vh.transpose(),Sinv,U.transpose()])\n",
    "Ap= Winv.dot(Mf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
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\"ticks\": \"\"}}, \"title\": {\"x\": 0.05}, \"xaxis\": {\"automargin\": true, \"gridcolor\": \"white\", \"linecolor\": \"white\", \"ticks\": \"\", \"title\": {\"standoff\": 15}, \"zerolinecolor\": \"white\", \"zerolinewidth\": 2}, \"yaxis\": {\"automargin\": true, \"gridcolor\": \"white\", \"linecolor\": \"white\", \"ticks\": \"\", \"title\": {\"standoff\": 15}, \"zerolinecolor\": \"white\", \"zerolinewidth\": 2}}}},\n",
       "                        {\"responsive\": true}\n",
       "                    ).then(function(){\n",
       "                            \n",
       "var gd = document.getElementById('b58ba72c-e2d0-43a6-b229-61559ec6e186');\n",
       "var x = new MutationObserver(function (mutations, observer) {{\n",
       "        var display = window.getComputedStyle(gd).display;\n",
       "        if (!display || display === 'none') {{\n",
       "            console.log([gd, 'removed!']);\n",
       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
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       "\n",
       "                        })\n",
       "                };\n",
       "                });\n",
       "            </script>\n",
       "        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fffig = go.Figure(data=go.Volume(x=Xf, y=Yf, z=Zf,value=Ap,\n",
    "        opacity=0.1,# needs to be small to see through all surfaces\n",
    "        surface_count=50 #needs to be a large number for good volume rendering\n",
    "        ))\n",
    "fffig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10000004223547876"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ap.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "App=np.linalg.pinv(W).dot(Mf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure()\n",
    "plt.imshow(rho,cmap='jet',aspect='equal')\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.linalg.cond(W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.close()"
   ]
  }
 ],
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